A steel rod of length 100 cm is clamped at the middle. The frequency of the fundamental mode for the longitudinal vibrations of the rod is
(Speed of sound in steel = 5 km `s^(-1)`) a) 1.5 KHz b) 2 KHz c) 2.5 KHz d) 3 KHz

Options

A. 1.5 KHz
B. 2 KHz
C. 2.5 KHz
D. 3 KHz

Explanation

To solve the problem of finding the frequency of the fundamental mode for the longitudinal vibrations of a steel rod clamped at the middle, we will follow these steps:

### Step 1: Convert the Length of the Rod
The length of the steel rod is given as 100 cm. We need to convert this into meters for consistency with the speed of sound.

\[ L = 100 \text{ cm} = \frac{100}{100} \text{ m} = 1 \text{ m} \]

### Step 2: Convert the Speed of Sound
The speed of sound in steel is given as 5 km/s. We will convert this into meters per second.

\[ V = 5 \text{ km/s} = 5 \times 1000 \text{ m/s} = 5000 \text{ m/s} \]

### Step 3: Identify the Mode of Vibration
Since the rod is clamped at the middle, it will have a node at the center and antinodes at both ends. For the fundamental mode, the distance between two consecutive antinodes is equal to half the wavelength (λ/2).

### Step 4: Relate Length to Wavelength
The length of the rod (L) is equal to half the wavelength (λ/2) because it has one full wave between the two antinodes at the ends. Therefore, we can write:

\[ \frac{\lambda}{2} = L \implies \lambda = 2L \]

Substituting the length of the rod:

\[ \lambda = 2 \times 1 \text{ m} = 2 \text{ m} \]

### Step 5: Calculate the Frequency
The frequency (ν) of the wave can be calculated using the formula:

\[ \nu = \frac{V}{\lambda} \]

Substituting the values we have:

\[ \nu = \frac{5000 \text{ m/s}}{2 \text{ m}} = 2500 \text{ Hz} \]

### Step 6: Convert Frequency to Kilohertz
Since 1 kHz = 1000 Hz, we can convert the frequency:

\[ \nu = 2500 \text{ Hz} = 2.5 \text{ kHz} \]

### Conclusion
The frequency of the fundamental mode for the longitudinal vibrations of the steel rod is **2.5 kHz**. Therefore, the correct answer is option **c) 2.5 kHz**.

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